In a non-coordinated, conventional cellular communication system, the transmissions to different users are formed independently. Hence, the transmission to one user typically acts as unwanted interference to other users. Because the system forms the transmission to each user independently, the system has no way of knowing how a transmission to a particular user will impact other users in the vicinity. As a result, with small to medium-sized cells and orthogonal multiple access within one cell, other-cell interference is a main factor limiting the performance of evolved cellular systems. Particularly for users near the cell edge, the other-cell interference is a main factor prohibiting the delivery of high data rate to these users.
In a non-coordinated multiple-input multiple output (MIMO) transmission with channel state information (CSI) available at the transmitter, where the transmitted signal for user u0 is formed independently of what interference this transmitted signal generates at other users, the output vector y received by the desired user u0 is given by Equation (1) belowy=Hx+n,  (1),where x is a vector of size t×1 that represents the total signal transmitted from all the transmitted antennas, H is a fixed channel response matrix for user u0 whose size is r×t (the channel matrix for the desired user is fixed and is perfectly known at the transmitter and by the desired user u0), and n is a Gaussian, circularly symmetric, complex-valued random noise vector with zero mean and covariance N0 I.
LetS=E[xx†]  (2)represent the transmit covariance matrix, and under the total average power constraint at the transmitter, we require thattr(S)≦P0.  (3)
The maximum data rate that can be reliably transmitted over the channel H is referred to as the capacity of the MIMO channel H. This capacity is obtained by maximizing the mutual information between x and y over all possible choices of S:
                              minimize          ⁢                                          ⁢                      Φ            ⁡                          (              S              )                                      =                              -            log                    ⁢                                          ⁢                      det            ⁡                          (                              I                +                                                      1                                          N                      0                                                        ⁢                                      HSH                    †                                                              )                                                          (        4        )                                          subject          ⁢                                          ⁢          to          ⁢                                          ⁢          S                ≥        0                            (        5        )                                          tr          ⁡                      (            S            )                          ≤                  P          0                                    (        6        )            
Viewed as an optimization problem over the set of positive semidefinite matrices of size t×t, denoted by S+t, the problem in Equation (4) is convex; since, the objective function
The Lagrangian associated with the optimization problem equation (4) is:
                              L          ⁡                      (                          S              ,                              l                1                            ,              Y                        )                          =                                            -              log                        ⁢                                                  ⁢                          det              (                              I                +                                                      1                                          N                      0                                                        ⁢                                      HSH                    †                                                              )                                +                                    l              1                        ⁡                          (                                                tr                  ⁡                                      (                    S                    )                                                  -                                  P                  0                                            )                                -                      tr            ⁡                          (              YS              )                                                          (        7        )            where λ1 is the Lagrangian multiplier associated with the constraint equation (5), and Ψ is the Lagrangian multiplier matrix associated with the constraint equation (3). Given that the problem is convex, and assuming that a strictly feasible S exists, the following KKT conditions are necessary and sufficient for optimality:
                                          ∂            L                                ∂            S                          =                                                            -                                  1                                      N                    0                                                              ⁢                                                                    H                    †                                    ⁡                                      (                                          I                      +                                                                        1                                                      N                            0                                                                          ⁢                                                  HSH                          †                                                                                      )                                                                    -                  1                                            ⁢              H                        +                                          λ                1                            ⁢              I                        -            Ψ                    =          0                                    (        8        )                                          tr          ⁡                      (                          Ψ              ⁢                                                          ⁢              S                        )                          =        0                            (        9        )                                                      λ            1                    ⁡                      (                                          tr                ⁡                                  (                  S                  )                                            -                              P                0                                      )                          =        0                            (        10        )                                          λ          1                ,                  Ψ          ≥          0                                    (        11        )            
By direct substitution, it can be easily verified that the well-known waterfilling solution for S satisfies the KKT conditions of equation (8):
                    H        =                  U          ⁢                                          ⁢          Λ          ⁢                                          ⁢                      V            †                                              (        12        )                                Λ        =                  diag          ⁡                      (                                          μ                1                            ,                              μ                2                            ,              …              ⁢                                                          ,                              μ                r                            ,              0              ,              …              ⁢                                                          ,              0                        )                                              (        13        )                                S        =                  V          ⁢                                          ⁢                      diag            ⁡                          (                                                P                  1                                ,                                  P                  2                                ,                …                ⁢                                                                  ,                                  P                  r                                ,                0                ,                …                ⁢                                                                  ,                0                            )                                ⁢                      V            †                                              (        14        )                                          P          i                =                              (                                          1                                  λ                  1                                            -                                                N                  0                                                  μ                  i                  2                                                      )                    +                                    (        15        )                                                      ∑                          i              =              1                        r                    ⁢                                    (                                                1                                      λ                    1                                                  -                                                      N                    0                                                        μ                    i                    2                                                              )                        +                          =                  P          0                                    (        16        )            
It is important to note that the solution for S obtained in equations (12-16) result in tr(S)=P0 and λ1>0, i.e. the obtained S satisfies the following specific version of the KKT conditions:
                                                        -                              1                                  N                  0                                                      ⁢                                                            H                  †                                ⁡                                  (                                      I                    +                                                                  1                                                  N                          0                                                                    ⁢                                              HSH                        †                                                                              )                                                            -                1                                      ⁢            H                    +                                    λ              1                        ⁢            I                    -          Ψ                =        0                            (        17        )                                          tr          ⁡                      (                          Ψ              ⁢                                                          ⁢              S                        )                          =        0                            (        18        )                                                      tr            ⁡                          (              S              )                                -                      P            0                          =        0                            (        19        )                                          λ          1                >        0                            (        20        )                                Ψ        ≥        0                            (        21        )            
A coordinated cellular communication system with distributed antennas uses it knowledge of the propagation environment to control the mutual interference by jointly shaping the signals that are transmitted to all the users. Coordinated transmission methods on the downlink must fundamentally adjust the transmission to each user so that two competing goals are satisfied: (1) a transmission to a user should be configured such that the signal corresponding to the transmission that is received at the antenna(s) of the user is strong and (2) the signal corresponding to the transmission that is received by the antennas of all other users should be weak.
Zero-Forcing Coordinated Transmission
With zero-forcing transmission, the transmission to each user generates absolutely no interference to any other active user in the coordinated multipoint (CoMP) cell, while the total power transmitted to each user must be below a pre-specified limit P0. See e.g., G. Caire and S. Shamai, “On the achievable throughput in multiantenna Gaussian broadcast channel,” IEEE Trans. Infor. Theory, vol. 49, July 2003.
Problem with Existing Technology
A problem with non-coordinated transmission is that it ignores completely the fact that in a wireless environment, the transmission to one user acts as interference to all other users. A problem with zero-forcing, coordinated transmission is that it breaks down at high system loads. CoMP methods on the downlink must fundamentally adjust the transmission to each user so that two competing goals are satisfied: (1) the signal received by the antennas of user u0 must be strong, (2) the signal received by the antennas of all other users must be weak. By making the interference zero, the zero-forcing method fundamentally puts too much emphasis on the second goal, at the expense of ignoring the first goal. At high loads, there are a large number of antennas that the transmitted signal for user u0 must be projected away from; hence, the transmitted signal is restricted to a very small sub-space. In this case, it is quite likely that transmitting in this very restricted sub-space will result in a small received signal power at the desired user u0. This effect is similar to the well-known “noise-enhancement” problem associated with a zero-forcing receiver.